MINISTRY OF EDUCATION AND VIETNAM ACADEMY
TRAINING OF SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY
Hoang Manh Tuan
DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS
FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS
Major: Applied Mathematics
Code: 9 46 01 12
SUMMARY OF DOCTORAL THESIS
HANOI - 2021
This thesis has been completed at
: Graduate University of Science and Technology – Vietnam Academy of
Science and Technology.
Supervisor 1: Prof. Dr. Dang Quang A
Supervisor 2: Assoc. Prof. Dr. Habil. Vu Hoang Linh
Reviewer 1:
Reviewer 2:
Reviewer 3:
The thesis will be defended at the Board of Examiners of Graduate University of Science and Technology –
Vietnam Academy of Science and Technology at ............................ on..............................
The thesis can be explored at:
- Library of Graduate University of Science and Technology
- National Library of Vietnam
INTRODUCTION
1. Overview of research situation
Many essential phenomena and processes arising in fields of science and technology are mathematically
modeled by ODEs of the form:
dy(t)
dt =fy(t),y(t0) = y0∈Rn,(0.0.1)
where
y(t)
denotes the vector-function
y1(t),y2(t),...,yn(t)T
, and the function
f
satisfies appropriate condi-
tions which guarantee that solutions of the problem
(0.0.1)
exist and are unique. The problem
(0.0.1)
is called
an initial value problem (IVP) or also a Cauchy problem.
The problem
(0.0.1)
has always been playing an essential role in both theory and practice. Theoretically,
it is not difficult to prove the existence, uniqueness and continuous dependence on initial data of the solutions of
the problem
(0.0.1)
thanks to the standard methods of mathematical analysis. However, it is very challenging,
even impossible, to solve the problem
(0.0.1)
exactly in general. In common real-world situations, the problem
of finding approximate solutions is almost inevitable. Consequently, the study of numerical methods for solving
ODEs has become one of the fundamental and practically important research challenges (see, for example,
Ascher and Petzold 1998; Burden and Faires 2011; Hairer, Nørsset and Wanner 1993, Hairer and Wanner 1996,
Stuart and Humphries 1998). Due to requirements of practice as well as the development of mathematical
theory, many numerical methods, typically finite difference methods have been constructed and developed. It is
safe to say that the general theory of the finite difference methods for the problem
(0.0.1)
has been developed
thoroughly in many monographs. These methods will be called the standard finite difference (SFD) methods to
distinguish them from the nonstandard finite difference (NSFD) schemes that will be presented in the remaining
parts.
Except for key requirements such as the convergence and stability, numerical schemes must correctly
preserve essential properties of corresponding differential equations. In other words, differential models must
be transformed into discrete models with the preservation of essential properties. However, in many problems,
the SFD schemes revealed a serious drawback called ”numerical instabilities”. To describe this, Mickens, the
creator of the concept of NSFD methods, wrote: ”numerical instabilities are an indication that the discrete
models are not able to model the correct mathematical properties of the solutions to the differential equations of
interest” (Mickens 1994, 2000, 2005, 2012). In a large number of works, Mickens discovered and analyzed
numerous examples related to the numerical instabilities occurring when using SFD methods. In 1980, Mickens
proposed the concept of NSFD schemes to overcome numerical instabilities. According to the Mickens’
methodology, NSFD schemes are those constructed following a set of basic rules derived from the analysis of
the numerical instabilities that occur when using SFD schemes (Mickens 1994, 2000, 2005, 2012).
Over the past four decades, the research direction on NSFD schemes has attracted the attention of
many researchers in many different aspects and gained a great number of interesting and significant results. All
of the works confirmed the usefulness and advantages of NSFD schemes. An advantage of NSFD schemes
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over standard ones is that they can correctly preserve essential properties (positivity, boundedness, asymptotic
stability, periodicity, etc.). In major surveys Mickens (2012) and Patidar (2005, 2016) and several monographs
Mickens (1994, 2000, 2005), Mickens and Padidar systematically presented results on NSFD methods in recent
decades as well as directions of the development in the future. Nowadays, NSFD methods have been and
will continue to be widely used as a powerful and effective approach to solve ODEs, PDEs, delay differential
equations (DDEs) and fractional differential equations (FDEs) (see, for instance, Arenas, Gonzalez-Parra and
Chen-Charpentier 2016; Garba et al. 2015; Ehrardt and Mickens 2013; Mickens 1994, 2000, 2005, 2012;
Modday, Hashim and Momani 2011; Patidar 2005, 2016).
2. The necessity of the research
Although the research direction on NSFD schemes for differential equations have achieved a lot of
results shown by both quantity and quality of existing research works, real-world situations have always posed
new complex problems in both qualitative study and numerical simulation aspects. On the other hand, there are
many differential models that have been established completely in the qualitative aspect but their corresponding
dynamically consistent discrete models have not yet been proposed and studied. Therefore, the construction of
discrete models that correctly preserve essential properties of differential models is truly necessary, has scientific
significance and needs to be studied. Importantly, the construction of NSFD schemes for ODE models still faces
many difficulties and has not been completely resolved, especially for models with at least one of the following
characteristics:
(i) Having higher dimensions.
(ii) Having non-hyberbolic equilibrium points.
(iii) Having global asymptotic stability (GAS) property.
Generally, most of existing results only focus on differential models having hyperbolic equilibrium points
with the local asymptotic stability (LAS), and there are no effective approaches for problems possessing non-
hyperbolic equilibrium points and/or having the GAS property. On the other hand, the study of the LAS of
NSFD schemes for models having higher dimensions is still a big challenge, and hence, effective approaches are
needed for these models. Furthermore, the improvement of the accuracy of NSFD schemes and the construction
of exact finite difference (EFD) schemes for ODE models are also essential with many important applications.
From the above reasons, we believe that the continuation of study on NSFD schemes for differential
equations is timely and really necessary and has great scientific and practical significance, and therefore, need
to be studied. That is why we set the aim of developing NSFD schemes for important mathematical models
modeled by ODEs, which arise in fields of science and technology.
3. Objectives and contents of the thesis
The aim of the thesis is to develop Mickens’ methodology to construct NSFD methods for solving some
important classes of differential equations arising in fields of science and technology.
The thesis intends to study the following contents:
Content 1: NSFD schemes for some classes of ODEs arising in science and technology.
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Content 2: EFD schemes for some ODEs and their applications.
Content 3: High order NSFD schemes for general autonomous dynamical systems and their applications.
4. Approach and research method
We will approach to the proposed contents of the thesis from both theoretical and practical points of
view. Differential models under consideration will be completely established on the qualitative aspect before
proposing and studying NSFD schemes. Numerical simulations will be also performed to confirm the validity
of theoretical results.
In order to perform the above research, we will use a combination of tools including qualitative
theory of discrete and continuous dynamical systems, the Lyapunov stability theory, the Mickens’ NSFD
methodology, theory of numerical methods and finite difference schemes for differential equations. In addition,
the experimental methods will be also used, especially when proofs for theoretical results are not finalized.
5. The new contributions of the thesis
1.
Proposing and analyzing NSFD schemes for some important classes of differential equations, which
are mathematical models of processes and phenomena arising in science and technology. The proposed
NSFD schemes are not only dynamically consistent with the differential equation models, but also easy
to be implemented; furthermore, they can be used to solve a large class of mathematical problems in both
theory and practice.
2.
Proposing novel efficient approaches and techniques to study asymptotic stability of the constructed
NSFD schemes.
3.
Constructing high-order NSFD methods for some classes of general dynamical systems; consequently,
the contradiction between dynamic consistency and high order of accuracy of NSFD methods has been
resolved.
4.
Proposing exact finite difference schemes for linear systems of differential equations with constant
coefficients. This result not only resolves some open questions related to exact schemes but also
generalizes some existing works.
5.
Performing many numerical experiments to confirm the theoretical results and to demonstrate the
advantages and superiority of the proposed NSFD schemes over the standard numerical schemes.
6. Structure of the thesis
In addition to ”Introduction”, ”General conclusions” and ”References”, the contents of this thesis are
presented in three chapters, among which the main results are in Chapters 2 and 3.
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